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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

How many numbers of 7 consecutive positive integers are divisible by 6?

(1) Their average is divisible by 6(2) Their median is divisible by 12

In the original condition, there is 1 variable since there are 7 consecutive positive integers. In order to match with the number of equations, you need 1 equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.For 1), there is always 1, which is unique and sufficient.For 2), there is always 1 as well, which is unique and sufficient.Therefore, the answer is D.

 For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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We're told that we have 7 CONSECUTIVE positive integers. We're asked how many of them are divisible by 6. To start, with 7 consecutive integers, only 1 or 2 of those integers will be divisible by 6. Once you have a number that is divisible by 6, the 'next' number that is divisible by 6 will either be '6 more' or '6 less' than the original number. For example: 0, 6, 12, 18, 24, etc.

1) Their average is divisible by 6.

With 7 consecutive integers, the AVERAGE will equal the "middle" number (in this case, the 4th number), so we can use a bit of 'brute force' to determine whether a pattern exists here or not. IF... the numbers are... 3, 4, 5, 6, 7, 8, 9 .... there's 1 number that is divisible by 6. 9, 10, 11, 12, 13, 14, 15 .... there's 1 number that is divisible by 6. 15, 16, 17, 18, 19, 20, 21 .... there's 1 number that is divisible by 6. Etc. There will ALWAYS be just 1 number that is divisible by 6. Fact 1 is SUFFICIENT

2) Their median is divisible by 12

With 7 consecutive integers, the AVERAGE = MEDIAN and both values will equal the "middle" number (in this case, the 4th number), so we can use a bit of 'brute force' here too. The difference is that the 4th value will have to be a multiple of 12. IF... the numbers are.... 9, 10, 11, 12, 13, 14, 15 .... there's 1 number that is divisible by 6. 21, 22, 23, 24, 25, 26, 27 .... there's 1 number that is divisible by 6. Etc. There will ALWAYS be just 1 number that is divisible by 6. Fact 2 is SUFFICIENT